the lambda calculus

David Walker

CS 441

the lambda calculus

• Originally, the lambda calculus was developed as a logic by Alonzo Church in 1932– Church says: “There may, indeed, be other

applications of the system than its use as a logic.”

– Dave says: “I’ll say”

Reading

• Pierce, Chapter 5

functions

• essentially every full-scale programming language has some notion of function– the (pure) lambda calculus is a language

composed entirely of functions– we use the lambda calculus to study the

essence of computation– it is just as fundamental as Turing Machines

syntax

t,e ::= x (a variable)

| \x.e (a function; in ML: fn x => e)

| e e (function application)

syntax

• the identity function:– \x.x

• 2 notational conventions:• applications associate to the left (like in ML): • “y z x” is “(y z) x”• the body of a lambda extends as far as possible to

the right:• “\x.x \z.x z x” is “\x.(x \z.(x z x))”

terminology

• \x.t

• \x.x y

the scope of x is the term t

x is boundin the term \x.x y

y is free in the term \x.x y

CBV operational semantics

• single-step, call-by-value OS: t --> t’– values are v ::= \x.t – primary rule (beta reduction):

– t [v/x] is the term in which all free occurrences of x in t are replaced with v

– this replacement operation is called substitution– we will define it carefully later in the lecture

(\x.t) v --> t [v/x]

operational semantics

• search rules:

• notice, evaluation is left to right

e1 --> e1’e1 e2 --> e1’ e2

e2 --> e2’v e2 --> v e2’

Example

(\x. x x) (\y. y)

Example

(\x. x x) (\y. y)

--> x x [\y. y / x]

Example

(\x. x x) (\y. y)

--> x x [\y. y / x]

== (\y. y) (\y. y)

Example

(\x. x x) (\y. y)

--> x x [\y. y / x]

== (\y. y) (\y. y)

--> y [\y. y / y]

Example

(\x. x x) (\y. y)

--> x x [\y. y / x]

== (\y. y) (\y. y)

--> y [\y. y / y]

== \y. y

Another example

(\x. x x) (\x. x x)

Another example

(\x. x x) (\x. x x)

--> x x [\x. x x/x]

Another example

(\x. x x) (\x. x x)

--> x x [\x. x x/x]

== (\x. x x) (\x. x x)

• In other words, it is simple to write non terminating computations in the lambda calculus

• what else can we do?

We can do everything

• The lambda calculus can be used as an “assembly language”

• We can show how to compile useful, high-level operations and language features into the lambda calculus– Result = adding high-level operations is

convenient for programmers, but not a computational necessity

– Result = make your compiler intermediate language simpler

Let Expressions

• It is useful to bind intermediate results of computations to variables:let x = e1 in e2

• Question: can we implement this idea in the lambda calculus?

source = lambda calculus + let

target = lambda calculus

translate/compile

Let Expressions

• It is useful to bind intermediate results of computations to variables:let x = e1 in e2

• Question: can we implement this idea in the lambda calculus?translate (let x = e1 in e2) =

(\x.e2) e1

Let Expressions

• It is useful to bind intermediate results of computations to variables:let x = e1 in e2

• Question: can we implement this idea in the lambda calculus?translate (let x = e1 in e2) =

(\x. translate e2) (translate e1)

Let Expressions

• It is useful to bind intermediate results of computations to variables:let x = e1 in e2

• Question: can we implement this idea in the lambda calculus?translate (let x = e1 in e2) =

(\x. translate e2) (translate e1)

translate (x) = x

translate (\x.e) = \x.translate e

translate (e1 e2) = (translate e1) (translate e2)

booleans

• we can encode booleans– we will represent “true” and “false” as functions

named “tru” and “fls”– how do we define these functions?– think about how “true” and “false” can be used– they can be used by a testing function:

• “test b then else” returns “then” if b is true and returns “else” if b is false

• the only thing the implementation of test is going to be able to do with b is to apply it

• the functions “tru” and “fls” must distinguish themselves when they are applied

booleans

• the encoding:

tru = \t.\f. t

fls = \t.\f. f

test = \x.\then.\else. x then else

booleans

tru = \t.\f. t fls = \t.\f. f

test = \x.\then.\else. x then else

eg:

test tru (\x.t1) (\x.t2)

-->* (\t.\f. t) (\x.t1) (\x.t2)

-->* \x.t1

booleans

tru = \t.\f. t fls = \t.\f. f

and = \b.\c. b c fls

and tru tru

-->* tru tru fls

-->* tru

booleans

tru = \t.\f. t fls = \t.\f. f

and = \b.\c. b c fls

and fls tru

-->* fls tru fls

-->* fls

booleans

• what is wrong with the following translation? translate true = tru

translate false = fls

translate (if e1 then e2 else e3)

= test (translate e1) (translate e2) (translate e3)

...

booleans

• what is wrong with the following translation? translate true = tru

translate false = fls

translate (if e1 then e2 else e3)

= test (translate e1) (translate e2) (translate e3)

...

-- e2 and e3 will both be evaluated regardlessof whether e1 is true or false-- the target program might not terminate in some cases when the source program would

pairs

• would like to encode the operations– create e1 e2– fst p– sec p

• pairs will be functions– when the function is used in the fst or sec

operation it should reveal its first or second component respectively

pairs

create = \fst.\sec.\bool. bool fst sec

fst = \p. p tru

sec = \p. p fls

fst (create tru fls)

-->* fst (\bool. bool tru fls)

-->* (\bool. bool tru fls) tru

-->* tru

and we can go on...

• numbers• arithmetic expressions (+, -, *,...)• lists, trees and datatypes• exceptions, loops, ...• ...• the general trick:

– values will be functions – construct these functions so that they return the appropriate information when called by an operation

Formal details

• In order to be precise about the operational semantics of the lambda calculus, we need to define substitution properly– remember the primary evaluation rule:

(\x.t) v --> t [v/x]

substitution: a first try

• the definition is given inductively:

x [t/x] = t

y [t/x] = y (if y ≠ x)

(\y.t’) [t/x] = \y.t’ [t/x]

t1 t2 [t/x] = (t1 [t/x]) (t2 [t/x])

substitution: a first try

• This works well 50% of the time• Fails miserably the rest of the time:

(\x. x) [y/x] = \x. y

• the x in the body of (\x. x) refers to the argument of the function

• a substitution should not replace the x• we got “unlucky” with our choice of variable names

substitution: a first try

• This works well 50% of the time• Fails miserably the rest of the time:

(\x. z) [x/z] = \x. x

• the z in the body of (\x. z) does not refer to the argument of the function

• after substitution, it does refer to the argument • we got “unlucky” with our choice of variable names

again!

calculating free variables

• To define substitution properly, we must be able to calculate the free variables precisely:

FV(x) = {x}

FV(\x.t) = FV(t) / {x}

FV(t1 t2) = FV(t1) U FV(t2)

substitution

x [t/x] = t y [t/x] = y (if y ≠ x)

(\y.t’) [t/x] = \y.t’ (if y = x)

(\y.t’) [t/x] = \y.t’ [t/x] (if y ≠ x and y FV(t))

t1 t2 [t/x] = (t1 [t/x]) (t2 [t/x])

substitution

• almost! But the definition is not exhaustive

• what if y ≠ x and y FV(t) in the case for functions:

(\y.t’) [t/x] = \y.t’ (if y = x)

(\y.t’) [t/x] = \y.t’ [t/x] (if y ≠ x and y FV(t))

alpha conversion

• the names of bound variables are unimportant (as far as the meaning of the computation goes)

• in ML, there is no difference between– fn x => x and fn y => y

• we will treat \x. x and \y. y as if they are (absolutely and totally) indistinguishable so we can always use one in place of the other

alpha conversion

• in general, we will adopt the convention that terms that differ only in the names of bound variables are interchangeable– ie: \x.t == \y. t[y/x] (where this is the latest

version of substitution)– changing the name of a bound variable is

called alpha conversion

substitution, finally

x [t/x] = t y [t/x] = y (if y ≠ x)

(\y.t’) [t/x] = \y.t’ [t/x] (if y ≠ x and y FV(t))

t1 t2 [t/x] = (t1 [t/x]) (t2 [t/x])

we use alpha-equivalent terms so this constraint can always be satisfied. We pick y as we likeso this is true.

operational semantics again

• Is this the only possible operational semantics?

e1 --> e1’e1 e2 --> e1’ e2

e2 --> e2’v e2 --> v e2’

(\x.t) v --> t [v/x]

alternatives

e1 --> e1’e1 e2 --> e1’ e2

e2 --> e2’v e2 --> v e2’

(\x.t) v --> t [v/x]

e1 --> e1’e1 e2 --> e1’ e2

(\x.t) e --> t [e/x]

call-by-value call-by-name

alternatives

e1 --> e1’e1 e2 --> e1’ e2

e2 --> e2’v e2 --> v e2’

(\x.t) v --> t [v/x]

e1 --> e1’e1 e2 --> e1’ e2

(\x.t) e --> t [e/x]

call-by-value full beta-reduction

e2 --> e2’e1 e2 --> e1 e2’

e --> e’\x.e --> \x.e’

alternatives

e1 --> e1’e1 e2 --> e1’ e2

e2 --> e2’v e2 --> v e2’

(\x.t) v --> t [v/x]

e1 --> e1’ e1 ≠ ve1 e2 --> e1’ e2

(\x.t) e --> t [e/x]

call-by-value normal-order reduction

e2 --> e2’v e2 --> v e2’

e --> e’\x.e --> \x.e’

alternatives

e1 --> e1’e1 e2 --> e1’ e2

e2 --> e2’v e2 --> v e2’

(\x.t) v --> t [v/x]

call-by-value right-to-left call-by-value

e1 --> e1’e1 v --> e1’ v

e2 --> e2’e1 e2 --> e1 e2’

(\x.t) v --> t [v/x]

summary

• the lambda calculus is a language of functions– Turing complete– easy to encode many high-level language

features

• the operational semantics– primary rule: beta-reduction– depends upon careful definition of substitution– many evaluation strategies